Introduction to Binomial Option Pricing

• The binomial option pricing model enables us to determine the price of an option, given the characteristics of the stock or other underlying asset
• The binomial option pricing model assumes that the price of the underlying asset follows a binomial distribution-that is, the asset price in each period can move only up or down by a specified amount
• The binomial model is often referred to as the “Cox-Ross-Rubinstein pricing model” (CRR)

A One-Period Binomial Tree

• Example
  • Consider a European call option on the stock of XYZ, with a \(\$40\) strike and 1 year to expiration
  • XYZ does not pay dividends, and its current price is \(\$41\)
  • The continuously compounded risk-free interest rate is \(8\%\)
  • The following figure depicts possible stock prices over 1 year, i.e., a binomial tree

Computing the Option Price

• Next, consider two portfolios
  • Portfolio A: buy one call option
  • Portfolio B: buy \(2/3\) shares of XYZ and borrow \(\$18.462\) at the risk-free rate

• Costs
  • Portfolio A: the call premium (this is unknown and what we are solving for)
  • Portfolio B: \(2/3 \times \$41 - \$18.462 = \$8.871\)

Computing the Option Price (cont’d)

Payoffs:

Portfolio A:

Portfolio B:

Computing the Option Price (cont’d)

• Portfolios A and B have the same payoff. Therefore
  • Portfolios A and B should have the same cost. Since Portfolio B costs \(\$8.871\), the price of one option must be \(\$8.871\)
  • There is a way to create the payoff to a call by buying shares and borrowing. Portfolio B is a synthetic call
  • One option has the risk of \(2/3\) shares. The value \(2/3\) is the delta (\(\Delta\)) of the option: the number of shares that replicates the option payoff

The Binomial Solution

• How do we find a replicating portfolio consisting of \(\Delta\) shares of stock and a dollar amount \(B\) in lending, such that the portfolio imitates the option whether the stock rises or falls?
  • Suppose that the stock has a continuous dividend yield of \(\delta\), which is reinvested in the stock. Thus, if you buy one share at time \(t\), at time \(t+h\) you will have \(e^{\delta h}\) shares
  • If the length of a period is \(h\), the interest factor per period is \(e^{r h}\)
  • \(uS\) denotes the stock price when the price goes up, and \(dS\) denotes the stock price when the price goes down

The Binomial Solution (cont’d)

• Note that \(u (d)\) in the stock price tree is interpreted as one plus the rate of capital gain (loss) on the stock if it goes up (down)

• The value of the replicating portfolio at time \(h\), with stock price \(S_{h}\), is

\[ \Delta S_{h} e^{\delta h} + e^{r h} B \]

The Binomial Solution (cont’d)

• At the prices \(S_{h} = uS\) and \(S_{h} = dS\), a successful replicating portfolio will satisfy

\[ \begin{aligned} & \left(\Delta \times uS \times e^{\delta h}\right)+\left(B \times e^{rh}\right)=C_{u} \\ & \left(\Delta \times dS \times e^{\delta h}\right)+\left(B \times e^{rh}\right)=C_{d} \end{aligned} \]

• Solving for \(\Delta\) and \(B\) gives \[ \begin{aligned} & \Delta = e^{-\delta \hbar} \frac{C_{u} - C_{d}}{S(u - d)} \\ & B = e^{-rh} \frac{uC_{d} - dC_{u}}{u - d} \end{aligned} \]

The Binomial Solution (cont’d)

• The cost of creating the option is the net cash flow required to buy the shares and bonds. Thus, the cost of the option is \(\Delta S+B\)

\[ \Delta S + B = e^{-rh} \left( C_{u} \frac{e^{(r-\delta) h} - d}{u - d} + C_{d} \frac{u - e^{(r-\delta)h}}{u - d} \right) \]

• The no-arbitrage condition is

\[ u > e^{(r-\delta)h} > d \]

Arbitraging a Mispriced Option

• If the observed option price differs from its theoretical price, arbitrage is possible
• If an option is overpriced, we can sell the option. However, the risk is that the option will be in the money at expiration, and we will be required to deliver the stock. To hedge this risk, we can buy a synthetic option at the same time we sell the actual option
• If an option is underpriced, we buy the option. To hedge the risk associated with the possibility of the stock price falling at expiration, we sell a synthetic option at the same time

A Graphical Interpretation of the Binomial Formula

• The portfolio describes a line with the formula

\[ C_{h} = \Delta S_{h} + e^{rh} B \]

• Where \(C_{h}\) and \(S_{h}\) are the option and stock value after one binomial period, and supposing \(\delta = 0\)
• We can control the slope of a payoff diagram by varying the number of shares, \(\Delta\), and its height by varying the number of bonds, \(B\)
• Any line replicating a call will have a positive slope ( \(\Delta > 0\) ) and negative intercept (\(B < 0\)). (For a put, \(\Delta < 0\) and \(B > 0\))

A Graphical Interpretation of the Binomial Formula (cont’d)

Risk-Neutral Pricing

• We can interpret the terms \(\left(e^{(r-\delta) h}-d\right) /(u-d)\) and \(\left(u-e^{(r-\delta) h}\right) /(u-d)\) as probabilities
• Let

\[ p^{\ast} = \frac{e^{(r - \delta)h} - d}{u - d} \]

• Then equation (10.3) can then be written as

\[ C = e^{-rh} \left[p^{\ast} C_{u} + \left(1 - p^{\ast} \right) C_{d} \right] \]

• We call \(p^{\ast}\) is the risk-neutral probability of an increase in the stock price

Summary

• In order to price an option, we need to know
• Stock price (\(S_{t}\))
• Strike price (\(K\))
• Standard deviation of returns on the stock (\(\sigma\))
• Dividend yield (\(\delta\))
• Risk-free rate (\(r\))
• Using the risk-free rate and \(\sigma\), we can approximate the future distribution of the stock by creating a binomial tree using equation (10.9)
• Once we have the binomial tree, it is possible to price the option using equation (10.3)

Continuously Compounded Returns

• Some important properties of continuously compounded returns
• The logarithmic function computes returns from prices

\[ \begin{aligned} r_{t, t+h} & = \ln\left(S_{t+h} / S_{t}\right) \\ & = \ln\left(S_{t+h}\right) - \ln\left(S_{t}\right) \end{aligned} \]

• The exponential function computes prices from returns

\[ S_{t+h} = S_{t} e^{r_{t,t+h}} \]

• Continuously compounded returns are additive

\[ r_{1,5} = r_{1,2} + r_{2,3} + r_{3,4} + r_{4,5} \]

Volatility

• Suppose the continuously compounded return over month \(i\) is \(r_{\text {monthly } i i}\). Since returns are additive, the annual return is

\[ r_{\text{amnual}} = \sum_{i=1}^{12} r_{\text{monthly}, i} \]

• The variance of the annual return is

\[ \operatorname{Var}\left(r_{\text{annual}}\right) = \operatorname{Var}\left(\sum_{i=1}^{12} \mathrm{r}_{\text{monthly},i}\right) \]

The Standard Deviation of Continuously Compounded Returns (cont’d)

• Suppose that returns are uncorrelated over time and that each month has the same variance of returns. Then from equation (10.13) we have

\[ \sigma^{2} = 12 \times \sigma^{2}_\text{(monthly)} \]

where \(\sigma^{2}\) denotes the annual variance

• Taking the square root of both sides yields

\[ \sigma=\sigma_{\text{monthly }} \sqrt{12} \]

• To generalize this formula, if we split the year into \(n\) periods of length \(h\) (so that \(h=1 / n\) ), the standard deviation over the period of length \((h, \sigma_{h})\) is

\[ \sigma = \frac{\sigma_{h}}{\sqrt{h}} \]

Constructing \(u\) and \(d\)

• In the absence of uncertainty, a stock must appreciate at the risk-free rate less the dividend yield. Thus, from time \(t\) to time \(t+h\), we have

\[ F_{t,t+h} = S_{t+h} = S_{t}e^{(r - \delta)h} \]

• The stock price next period equals the forward price (w/o uncertainty!)

Constructing \(u\) and \(d\) (cont’d)

• With uncertainty, the stock price evolution is

\[ \begin{aligned} uS_{t} & = F_{t,t+h} e^{+\sigma\sqrt{h}} \\ dS_{t} & = F_{t,t+h} e^{-\sigma\sqrt{h}} \end{aligned} \]

• Where \(\sigma\) is the annualized standard deviation of the continuously compounded return, and \(\sigma \sqrt{h}\) is standard deviation over a period of length \(h\)

• We can also rewrite (10.17) as

\[ \begin{aligned} u & =e^{(r - \delta)h + \sigma \sqrt{h}} \\ d & =e^{(r - \delta)h - \sigma \sqrt{h}} \end{aligned} \]

• We refer to a tree constructed using equation (10.9) as a forward tree

Estimating Historical Volatility

• We need to decide what value to assign to \(\sigma\), which we cannot observe directly

• One possibility is to measure \(\sigma\) by computing the standard deviation of continuously compounded historical returns

  • Volatility computed from historical stock returns is historical volatility
  • For example, calculate the standard deviation with weekly data, then annualize the result by using equation (10.15)
  • For standard options, the volatility that matters excludes dividends

Estimating Historical Volatility (cont’d)

One-Period Example with a Forward Tree

• Consider a European call option on a stock, with a \(\$40\) strike and 1 year to expiration. The stock does not pay dividends, and its current price is \(\$ 41\). Suppose the volatility of the stock is \(30\%\)
• The continuously compounded risk-free interest rate is \(8\%\)
• \(\mathrm{S} = 41, \mathrm{r} = 0.08, \delta = 0, \sigma = 0.30\), and \(\mathrm{h} = 1\)
• Use these inputs to
  • calculate the final stock prices
  • calculate the final option values
  • calculate \(\Delta\) and B
  • calculate the option price

One-Period Example with a Forward Tree (cont’d)

• Calculate the final stock prices

\[ \begin{aligned} uS &= \$41 e^{(0.08-0) \times 1 + 0.3 \times \sqrt{1}} = \$59.954 \\ dS &= \$41 e^{(0.08-0) \times 1 - 0.3 \times \sqrt{1}} = \$32.903 \end{aligned} \quad \Rightarrow \quad \begin{aligned} & u = \frac{\$59.954}{S} = \frac{59.954}{41} = 1.4623 \\ & d = \frac{\$32.903}{S} = \frac{32.903}{41} = 0.8025 \end{aligned} \]

• Calculate the final option values

\[ \begin{aligned} C_{u} &= \max(uS - K, 0) =\max(59.954 - 40, 0) = 19.954 \\ C_{d} &= \max(dS - K, 0) =\max(32.903 - 40, 0) = 0 \end{aligned} \]

One-Period Example with a Forward Tree (cont’d)

• Calculate \(\Delta\) and \(B\)

\[ \begin{aligned} \Delta &= \left(\frac{19.954-0}{\$41 \times (1.4632 - 0.8025)} \right)= 0.7376 \\ B &= e^{-0.08} \left(\frac{1.4632 \times \$0 - 0.8025 \times \$19.954}{1.4632 - 0.8025} \right) = -\$22.405 \end{aligned} \]

• Calculate the option price

\[ \Delta S + B = 0.7376 \times 41 - \$22.405 = \$7.839 \]

One-Period Example with a Forward Tree (cont’d)